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Publication numberUS7486418 B2
Publication typeGrant
Application numberUS 11/377,161
Publication dateFeb 3, 2009
Filing dateMar 16, 2006
Priority dateDec 28, 2005
Fee statusPaid
Also published asUS20070146748
Publication number11377161, 377161, US 7486418 B2, US 7486418B2, US-B2-7486418, US7486418 B2, US7486418B2
InventorsSergey N. Bezryadin
Original AssigneeKwe International, Inc.
Export CitationBiBTeX, EndNote, RefMan
External Links: USPTO, USPTO Assignment, Espacenet
Editing (including contrast and sharpness editing) of digital images
US 7486418 B2
Abstract
In a digital image having pixels pi (i=1,2, . . . ) and a brightness B(pi) at each pixel pi, the contrast editing is performed by replacing the brightness B(pi) with B*(pi)=Bavg 1−ε(pi)×Bo ε(pi), where ε is a positive constant other than 1, and Bavg is a weighted average of the B values in an image region R(pi) containing the pixel pi. The image can be a color image, with color represented using digital values B (brightness), e and f such that B=√{square root over (D2+E2+F2)}, e=E/B, f=F/B, where DEF is a linear color coordinate system. Alternatively, color can be represented using digital values B, C (chroma) and H (hue), where cos C=D/B and tan H=E/F. The contrast can be edited without a color shift (i.e. without changing the chromaticity coordinates) by changing the B coordinate and leaving unchanged the other coordinates e and f or C and H. Sharpness can be edited using the same techniques with a small region R(pi).
Images(4)
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Claims(34)
1. A circuitry-implemented method comprising image editing, the method comprising:
(1) obtaining digital data for image portions p1, p2, . . . , wherein for each image portion p (p=p1, p2, . . . ), the digital data represent coordinates of a color S(p) of the portion p in a first coordinate system, wherein for a color S having tristimulus values T1, T2, T3 in a second coordinate system, the coordinates of the color S in the first coordinate system are coordinates S1, S2, S3, or a linear transformation of the coordinates S1, S2, S3, wherein:
(A) the coordinate S1 is defined by a B value

B=√{square root over (g11 T 1 2 +g 22 T 2 2 +g 33 T 3 2 +g 12 T 1 T 2 +g 13 T 1 T 3 +g 23 T 2 T 3)}
wherein g11, g22, g33, g12, g13, g23 are predefined constants, and g11, g22, g33 are not equal to zero, or
(B) the coordinate S1 is defined by the B value and by a sign of a predefined function of one or more of T1, T2, T3;
(2) for at least one image portion pi which is one of p1, p2, . . . and whose respective color S(pi) has tristimulus values T1=T1(pi), T2=T2(pi), T3=T3(pi) in the second coordinate system and has a B value B(pi), obtaining color coordinates in the first color coordinate system of a modified color S*(pi) which has tristimulus values T1=T1*(pi), T2=T2*(pi), T3=T3*(pi) in the second coordinate system and has a B value B*(pi) such that:
B * ( p i ) = B avg ( p i ) × f ( B ( p i ) B avg ( p i ) )
wherein:
ƒ is a predefined strictly increasing non-identity function; and
Bavg(pi) is a function of the B values of image portions in an image region R(pi) containing a plurality of image portions including the portion pi.
2. The method of claim 1 wherein Bmin(pi)≦Bavg(pi)≦Bmax(pi), wherein Bmin(pi) is the minimum of the B values in the region R(pi), and Bmax(pi) is the maximum of the B values in the region R(pi).
3. The method of claim 2 wherein the operation (2) is performed for a plurality of portions pi with each image region R(pi) having a predefined geometry with respect to the respective portion pi.
4. The method of claim 1 wherein the image region R(pi) contains all said image portions.
5. The method of claim 1 wherein each image portion is a pixel, and the image region R(pi) contains at most 30% of said pixels.
6. The method of claim 5 wherein the image region R(pi) contains at least 10% of said pixels.
7. The method of claim 1 wherein each image portion is a pixel, and the image region R(pi) contains at most 1% of said pixels.
8. The method of claim 1 wherein each image portion is a pixel, and the region R(pi) is contained in a rectangle of at most 31 pixels by 31 pixels, the rectangle being centered at the pixel pi.
9. The method of claim 1 wherein each image portion is a pixel, and the method further comprises:
receiving a command to edit an image comprising said image portions;
determining if the command is a command of a first type or a command of a second type;
if the command is a command of a first type, then performing the operation (2) with the image region R(pi) being contained in a rectangle of at most 31 pixels by 31 pixels, the rectangle being centered at the pixel pi;
if the command is of a second type, than performing the operation (2) with the image region R(pi) comprising at least 10% of the pixels.
10. The method of claim 1 wherein the command of the first type is a command to change a sharpness of at least a portion of the image, and the command of the second type is a command to change the image's global or local contrast.
11. The method of claim 1 wherein Bavg is a weighted average of the B values of the image region R(pi), and the sum of the weights is equal to 1.
12. The method of claim 1 wherein values T2/B and T3/B for the color S(pi) are the same as for the color S*(pi).
13. The method of claim 12 wherein a value T1/B for the color S(pi) is the same as for the color S*(pi).
14. The method of claim 1 wherein:

B=√{square root over (α1 2(T 1 ,T 2 ,T 3)+α2 2(T 1 ,T 2 ,T 3)+α3 2(T1,T2,T3))}{square root over (α1 2(T 1 ,T 2 ,T 3)+α2 2(T 1 ,T 2 ,T 3)+α3 2(T1,T2,T3))}{square root over (α1 2(T 1 ,T 2 ,T 3)+α2 2(T 1 ,T 2 ,T 3)+α3 2(T1,T2,T3))}
wherein

α1(T 1 ,T 2 ,T 3)=α11 ×T 112 ×T 213 ×T 3

α2(T 1 ,T 2 ,T 3)=α21 ×T 122 ×T 223 ×T 3

α3(T 1 ,T 2 ,T 3)=α31 ×T 132 ×T 233 ×T 3
wherein α11, α12, α13, α21, α22, α23, α31, α32, α33 are predefined numbers such that the following matrix Λ is non-degenerate:
Λ = [ α 11 α 12 α 13 α 21 α 22 α 23 α 31 α 32 α 33 ] .
15. The method of claim 14 wherein α1(T1,T2,T3), α2(T1,T2,T3), α3(T1,T2,T3) are tristimulus values corresponding to 70%-orthonormal color matching functions.
16. The method of claim 15 wherein α1(T1,T2,T3), α2(T1,T2,T3), α3(T1,T2,T3) are tristimulus values corresponding to 90%-orthonormal color matching functions.
17. The method of claim 15 wherein:
the value T1 is one of values D, E, F, the value T2 is another one of D, F, F, and the value T3 is the third one of D, F, F, where
[ D E F ] = A [ X Y Z ]
where the matrix A has elements which, up to rounding, are as follows:
A = [ 0.205306 0.712507 0.467031 - 0.365451 1.011998 - 0.610425 1.853667 - 1.279659 - 0.442859 ]
where X, Y, Z are the coordinates of the color S in the CIE 1931 XYZ color coordinate system for a 2° field;
wherein up to a constant multiple, Λ is a 70%-orthonormal matrix.
18. Circuitry for performing the method of claim 1.
19. One or more computer-readable mediums comprising computer instructions to cause a computer system to perform the method of claim 1.
20. A circuitry-implemented method comprising image editing, the method comprising:
(1) obtaining digital data for image portions p1, p2 . . . , wherein for each image portion p (p=p1,p2, . . . ), the digital data represent a brightness B(p) of the portion p;
(2) for at least one image portion pi, which is one of p1,p2, . . . , obtaining a brightness B* of a modified image, such that:
B * ( p i ) = B avg ( p i ) × f ( B ( p i ) B avg ( p i ) )
wherein:
ƒ is a predefined strictly increasing non-identity function; and
Bavg(pi) is a function of the brightness values B(pj) of image portions pj in an image region R(pi) containing the portion p1.
21. The method of claim 20 wherein the operation (2) is performed for a plurality of portions pi, with each image region R(pi) having a predefined geometry with respect to the respective portion pi.
22. The method of claim 20 wherein the image region R(pi) contains all said image portions.
23. The method of claim 20 wherein each image portion is a pixel, and the image region R(pi) contains at most 30% of said pixels.
24. The method of claim 23 wherein the image region R(pi) contains at least 10% of said pixels.
25. The method of claim 20 wherein each image portion is a pixel, and the image region R(pi) contains at most 1% of said pixels.
26. The method of claim 20 wherein each image portion is a pixel, and the region R(pi) is contained in a rectangle of at most 31 pixels by 31 pixels, the rectangle being centered at the pixel pi.
27. The method of claim 20 wherein each image portion is a pixel, and the method further comprises:
receiving a command to edit an image comprising said image portions;
determining if the command is a command of a first type or a command of a second type;
if the command is a command of a first type, then performing the operation (2) with the image region R(pi) being contained in a rectangle of at most 31 pixels by 31 pixels, the rectangle being centered at the pixel pi;
if the command is of a second type, than performing the operation (2) with the image region R(pi) comprising at least 10% of the pixels.
28. The method of claim 20 wherein the command of the first type is a command to change a sharpness of at least a portion of the image, and the command of the second type is a command to change the image's global or local contrast.
29. Circuitry for performing the method of claim 20.
30. One or more computer-readable mediums comprising computer instructions to cause a computer system to perform the method of claim 20.
31. The method of claim 1 wherein

B*(p i)=B avg 1−ε(p iB ε(p i)
wherein ε a positive constant other than 1, and Bavg is the mean of the B values of the image region R(pi).
32. The method of claim 20 wherein

B*(p i)=B avg 1−ε(p iB ε(p i)
wherein ε is a positive constant other than 1, and Bavg is a weighted average of the brightness values of the image region R(pi).
33. A data transmission method comprising transmitting a computer program over a network link, wherein the computer program is operable to cause a computer system to perform the method of claim 1.
34. A data transmission method comprising transmitting a computer program over a network link, wherein the computer program is operable to cause a computer system to perform the method of claim 20.
Description
CROSS REFERENCE TO RELATED APPLICATIONS

The present application is a continuation-in-part of U.S. patent application Ser. No. 11/321,443, filed Dec. 28, 2005 by Sergey N. Bezryadin, entitled “COLOR EDITING (INCLUDING BRIGHTNESS EDITING) USING COLOR COORDINATE SYSTEMS INCLUDING SYSTEMS WITH A COORDINATE DEFINED BY A SQUARE ROOT OF A QUADRATIC POLYNOMIAL IN TRISTIMULUS VALUES AND, POSSIBLY, BY A SIGN OF A FUNCTION OF ONE OR MORE OF TRISTIMULUS VALUES”, incorporated herein by reference. The present application is also a continuation-in-part of U.S. patent application Ser. No. 11/322,111, filed Dec. 28, 2005 by Sergey N. Bezryadin, entitled “COLOR COORDINATE SYSTEMS INCLUDING SYSTEMS WITH A COORDINATE DEFINED BY A SQUARE ROOT OF A QUADRATIC POLYNOMIAL IN TRISTIMULUS VALUES AND, POSSIBLY, BY A SIGN OF A FUNCTION OF ONE OR MORE OF TRISTIMULUS VALUES”, incorporated herein by reference.

BACKGROUND OF THE INVENTION

The present invention relates to editing of digital images, including both color and monochromatic images. A digital representation of an image can be stored in a storage device (e.g. a computer memory, a digital video recorder, or some other device). Such representation can be transmitted over a network, and can be used to display the image on a computer monitor, a television screen, a printer, or some other device. The image can be edited using a suitable computer program.

Color is a sensation caused by electromagnetic radiation (light) entering a human eye. The light causing the color sensation is called “color stimulus”. Color depends on the radiant power and spectral composition of the color stimulus, but different stimuli can cause the same color sensation. Therefore, a large number of colors can be reproduced (“matched”) by mixing just three “primary” color stimuli, e.g. a Red, a Blue and a Green. The primary stimuli can be produced by three “primary” light beams which, when mixed and reflected from an ideal diffuse surface, produce a desired color. The color can be represented by its coordinates, which specify the intensities of the primary light beams. For example, in linear RGB color coordinate systems, a color S is represented by coordinates R, G, B which define the intensities of the respective Red, Green and Blue primary light beams needed to match the color S. If P(λ) is the radiance (i.e. the energy per unit of time per unit wavelength) of a light source generating the color S, then the RGB coordinates can be computed as:

R = 0 P ( λ ) r _ ( λ ) λ G = 0 P ( λ ) g _ ( λ ) λ B = 0 P ( λ ) b _ ( λ ) λ ( 1 )
where r(λ), g(λ), b(λ) are “color matching functions” (CMF's). For each fixed wavelength λ, the values r(λ), g(λ), b(λ) are respectively the R, G and B values needed to match color produced by a monochromatic light of the wavelength λ of a unit radiance. The color matching functions are zero outside of the visible range of the λ values, so the integration limits in (1) can be replaced with the limits of the visible range. The integrals in (1) can be replaced with sums if the radiance P(λ) is specified as power at discrete wavelengths. FIG. 1 illustrates the color matching functions for the 1931 CIE RGB color coordinate system for a 2° field. (CIE stands for “Commission Internationale de l'Eclairage”.) See D. Malacara, “Color Vision and Colorimetry: theory and applications” (2002), and Wyszecki & Stiles, “Color Science: concepts and Methods, Quantitative Data and Formulae” (2nd Ed. 2000), both incorporated herein by reference.

The RGB system of FIG. 1 is called linear because, as shown by equations (1), the R, G, and B values are linear in P(λ). In a linear system, the intensities such as R, G, B are called “tristimulus values”.

As seen from FIG. 1, the function r(λ) can be negative, so the R coordinate can be negative. If R is negative, this means that when the color S is mixed with |R| units of the Red primary, the resulting color matches the mixture of G units of the Green primary with B units of the Blue primary.

New linear color coordinate systems can be obtained as non-degenerate linear transformations of other systems. For example, the 1931 CIE XYZ color coordinate system for a 2° field is obtained from the CIE RGB system of FIG. 1 using the following transformation:

[ X Y Z ] = A RGB - XYZ [ R G B ] where : ( 2 ) A RGB - XYZ = 1 0.17697 [ 0.49 0.31 0.20 0.17697 0.81240 0.01063 0.00 0.01 0.99 ] ( 3 )
This XYZ system does not correspond to real, physical primaries. The color matching functions x(λ), y(λ), z(λ) for this XYZ system are shown in FIG. 2. These color matching functions are defined by the same matrix ARGB-XYZ:

[ x _ ( λ ) y _ ( λ ) z _ ( λ ) ] = A RGB - XYZ [ r _ ( λ ) g _ ( λ ) b _ ( λ ) ]
The tristimulus values X, Y, Z can be computed from the color matching functions in the usual way:

X = 0 P ( λ ) x _ ( λ ) λ Y = 0 P ( λ ) y _ ( λ ) λ Z = 0 P ( λ ) z _ ( λ ) λ ( 4 )

There are also non-linear color coordinate systems. One example is a non-linear sRGB system standardized by International Electrotechnical Commission (IEC) as IEC 61966-2-1. The sRGB coordinates can be converted to the XYZ coordinates (4) or the CIE RGB coordinates (1). Another example is HSB (Hue, Saturation, Brightness). The HSB system is based on sRGB. In the HSB system, the colors can be visualized as points of a vertical cylinder. The Hue coordinate is an angle on the cylinder's horizontal circular cross section. The pure Red color corresponds to Hue=0°; the pure Green to Hue=120°; the pure Blue to Hue=240°. The angles between 0° and 120° correspond to mixtures of the Red and the Green; the angles between 120° and 240° correspond to mixtures of the Green and the Blue; the angles between 240° and 360° correspond to mixtures of the Red and the Blue. The radial distance from the center indicates the color's Saturation, i.e. the amount of White (White means here that R=G=B). At the circumference, the Saturation is maximal, which means that the White amount is 0 (this means that at least one of the R, G, and B coordinates is 0). At the center, the Saturation is 0 because the center represents the White color (R=G=B). The Brightness is measured along the vertical axis of the cylinder, and is defined as max(R,G,B).

Different color coordinate systems are suitable for different purposes. For example, the sRGB system is convenient for rendering color on certain types of monitors which recognize the sRGB coordinates and automatically convert these coordinates into color. The HSB system is convenient for some color editing operations including brightness adjustments.

Brightness can be thought of as a degree of intensity of a color stimulus. Brightness corresponds to our sensation of an object being “bright” or “dim”. Brightness has been represented as the Y value of the XYZ system of FIG. 2, or as the maximum of the R, G and B coordinates of the sRGB coordinate system. Other representations also exist. The saturation can be thought of as a measure of the white amount in the color.

Contrast can be thought of as the brightness difference between the brightest and the dimmest portions of the image or part of the image.

Exemplary contrast editing techniques are described in William K. Pratt, “DIGITAL IMAGE PROCESSING” (3ed. 2001), pages 243-252, incorporated herein by reference.

Sharpness relates to object boundary definition. The image is sharp if the object boundaries are well defined. The image is blurry if it is not sharp.

It is desirable to obtain color coordinate systems which facilitate contrast editing and other types of image editing of color and monochromatic images.

SUMMARY

This section summarizes some features of the invention. The invention is not limited to these features. The invention is defined by the appended claims.

In some embodiments, the contrast is edited as follows. Let us suppose that a digital image consist of a number of pixels. Let us enumerate these pixels as p1, p2, . . . (The invention is not limited to any pixel enumeration.) Let B(pi) denote the brightness at the pixel pi (i=1,2, . . . ). The contrast editing can be performed by changing the brightness B(pi) to the value
B*(p i)=B avg 1−ε(p iB ε(p i)
where ε is a positive constant other than 1, and Bavg is a weighted average (e.g. the mean) of the brightness values B in an image region R(pi) containing the pixel pi. Of note, this equation means that:

B * ( p i ) B avg ( p i ) = ( B ( p i ) B avg ( p i ) ) ɛ , or B * ( p i ) = B avg ( p i ) × ( B ( p i ) B avg ( p i ) ) ɛ

The invention is not limited to such embodiments. For example, the brightness can be edited according to the equation:

B * ( p i ) = B avg ( p i ) × f ( B ( p i ) B avg ( p i ) )
where ƒ is a predefined strictly increasing non-identity function, and Bavg(pi) is a function of the B values of image portions in an image region R(pi) containing the portion pi.

In some embodiments, the image is a color image. The color coordinate system is chosen so that the brightness B is one of the coordinates which can be changed without changing the chromaticity coordinates. For a linear color coordinate system with coordinates T1, T2, T3, the chromaticity coordinates are defined as:
T 1/(T 1 +T 2 +T 3), T 2/(T 1 +T 2 +T 3), T 3/(T 1 +T 2 +T 3).
In some embodiments, a non-linear color coordinate system is used in which one of the coordinates is defined by a square root of a quadratic polynomial in tristimulus values, e.g. by a value B=√{square root over (T1 2+T2 2+T3 2)} where T1, T2, T3 are tristimulus values, or by the value B and the sign of one or more of T1, T2, T3.

In some embodiments, the color coordinate system has the following coordinates:
B=√{square root over (T 1 2 +T 2 2 +T 3 2)}
S 2 =T 2 /B
S 3 =T 3 /B
In this coordinate system, if the B coordinate is changed, e.g. multiplied by some number k, and S2 and S3 are unchanged, the color modification corresponds to multiplying the tristimulus values by k. Such color modification does not change the chromaticity coordinates
T 1/(T 1 +T 2 +T 3), T 2/(T 1 +T 2 +T 3), T 3/(T 1 +T 2 +T 3).
Therefore, the (B, S2, S3) coordinate system facilitates color editing when it is desired not to change the chromaticity coordinates (so that no color shift would occur). Of note, there is a known color coordinate system xyY which also allows changing only one coordinate Y without changing the chromaticity coordinates. The xyY system is defined from the XYZ coordinates of FIG. 2 as follows:

the Y coordinate of the xyY system is the same as the Y coordinate of the XYZ system;
x=X/(X+Y+Z);
z=Z/(X+Y+Z).
In the xyY system, if Y is changed but x and y remain unchanged, then the chromaticity coordinates are unchanged. The xyY system differs from the (B, S2, S3) system in that the Y coordinate is a linear coordinate and is a tristimulus value, while the B coordinate is a non-linear function of tristimulus values (and of the power distribution P(λ)).

Linear transformations of such color coordinate systems can be used to obtain other novel color coordinate systems.

The techniques described above can be used to adjust either the global contrast, i.e. when the region R(pi) is the whole image, or the local contrast, when the region R(pi) is only a part of the image. In some embodiments, the region R(pi) contains 10% to 30% of the image in the local contrast adjustments, and other percentages are possible.

The inventor has observed that the same techniques can be used to change the image sharpness if the region R(pi) is small, e.g. 1% of the image or less. In some embodiments, the region R(pi) is contained in a rectangle of at most 31 pixels by 31 pixels, with the center at pixel pi. Even smaller outer rectangles can be used, e.g. 21×21 pixels, or 11×11 pixels, and other rectangles. The image may contain thousands, millions, or some other number of pixels.

The invention is not limited to the features and advantages described above. Other features are described below. The invention is defined by the appended claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIGS. 1 and 2 are graphs of color matching functions for prior art color coordinate systems.

FIG. 3 illustrates some color coordinate systems according to some embodiments of the present invention.

FIGS. 4 and 5 are block diagrams illustrating color editing according to some embodiments of the present invention.

FIG. 6 illustrates an image edited according to some embodiments of the present invention.

FIG. 7 illustrates an image extension at the image boundaries for contrast editing according to some embodiments of the present invention.

FIGS. 8, 9 are brightness graphs illustrating contrast editing according to some embodiments of the present invention.

FIG. 10 is a flowchart of an image editing method according to some embodiments of the present invention.

DESCRIPTION OF SOME EMBODIMENTS

The embodiments described in this section illustrate but do not limit the invention. The invention is defined by the appended claims.

Some embodiments of the present invention use color coordinate systems Bef and BCH which can be defined, for example, as follows. First, a linear color coordinate system DEF is defined as a linear transformation of the 1931 CIE XYZ color coordinate system of FIG. 2:

[ D E F ] = A XYZ - DEF [ X Y Z ] where : ( 5 ) A XYZ - DEF = [ 0.205306 0.712507 0.467031 1.853667 - 1.279659 - 0.442859 - 0.365451 1.011998 - 0.610425 ] ( 6 )
It has been found that for many computations, adequate results are achieved if the elements of matrix AXYZ-DEF are rounded to four digits or fewer after the decimal point, i.e. the matrix elements can be computed with an error Err≦0.00005. Larger errors can also be tolerated in some embodiments. The DEF coordinate system corresponds to color matching functions d(λ), ē(λ), f(λ) which can be obtained from x(λ), y(λ), z(λ) using the same matrix AXYZ-DEF:

[ d _ ( λ ) e _ ( λ ) f _ ( λ ) ] = A XYZ - DEF [ x _ ( λ ) y _ ( λ ) z _ ( λ ) ] ( 7 )
By definition of the color matching functions and the tristimulus values,

D = 0 P ( λ ) d _ ( λ ) λ E = 0 P ( λ ) e _ ( λ ) λ F = 0 P ( λ ) f _ ( λ ) λ ( 8 )
As explained above, the integration can be performed over the visible range only.

As seen in FIG. 2, the color matching functions x(λ), y(λ), z(λ) are never negative. It follows from equations (4) that the X, Y, Z values are never negative. Since the first row of matrix AXYZ-DEF has only positive coefficients, the function d(λ) is never negative, and the D value is also never negative.

When D>0 and E=F=0, the color is white or a shade of gray. Such colors coincide, up to a constant multiple, with the CIE D65 white color standard.

If a color is produced by a monochromatic radiation with λ=700 nm (this is a red color), then F=0 and E>0.

The color matching functions d(λ), ē(λ), f(λ) form an orthonormal system in the function space L2 on [0,∞) (or on any interval containing the visible range of the λ values if the color matching functions are zero outside of this range), that is:

0 d _ ( λ ) e _ ( λ ) λ = 0 d _ ( λ ) f _ ( λ ) λ = 0 e _ ( λ ) f _ ( λ ) λ = 0 0 [ d _ ( λ ) ] 2 λ = 0 [ e _ ( λ ) ] 2 λ = 0 [ f _ ( λ ) ] 2 λ = K ( 9 )
where K is a positive constant defined by the measurement units for the wavelength λ and the radiance P(λ). The units can be chosen so that K=1.

The integrals in (9) can be replaced with sums if the CMF's are defined at discrete λ values, i.e.:

λ d _ ( λ ) e _ ( λ ) = λ d _ ( λ ) f _ ( λ ) = λ e _ ( λ ) f _ ( λ ) = 0 λ [ d _ ( λ ) ] 2 = λ [ e _ ( λ ) ] 2 = λ [ f _ ( λ ) ] 2 = K ( 10 )
where the sums are taken over a discrete set of the λ values. The constant K can be different than in (9). Color matching functions will be called orthonormal herein if the satisfy the equations (9) or (10).

If S1 and S2 are two colors with DEF coordinates (D1,E1,F1) and (D2,E2,F2), a dot product of these colors can be defined as follows:
<S1,S2>=D1*D2+E1*E2+F1*F2  (11)
Thus, the DEF coordinate system can be thought of as a Cartesian coordinate system having mutually orthogonal axes D, E, F (FIG. 3), with the same measurement unit for each of these axes.

The dot product (11) does not depend on the color coordinate system as long as the color coordinate system is orthonormal in the sense of equations (9) or (10) and its CMF's are linear combinations of x(λ), y(λ), z(λ) (and hence of d(λ), ē(λ), f(λ)). More particularly, let T1, T2, T3 be tristimulus values in a color coordinate system whose CMF's t 1, t 2, t 3 belong to a linear span Span( d(λ), ē(λ), f(λ)) and satisfy the conditions (9) or (10). For the case of equations (9), this means that:

0 t 1 _ ( λ ) t 2 _ ( λ ) λ = 0 t 1 _ ( λ ) t 3 _ ( λ ) λ = 0 t 2 _ ( λ ) t 3 _ ( λ ) λ = 0 0 [ t 1 _ ( λ ) ] 2 λ = 0 [ t 2 _ ( λ ) ] 2 λ = 0 [ t 3 _ ( λ ) ] 2 λ = K ( 12 )
with the same constant K as in (9). The discrete case (10) is similar. Suppose the colors S1, S2 have the T1T2T3 coordinates (T1.1, T2.1, T3.1) and (T1.2, T2.2, T3.2) respectively. Then the dot product (11) is the same as
<S 1 ,S 2 >=T 1.1 T 1.2 +T 2.1 T 2.2 +T 3.1 T 3.2.

The brightness B of a color S can be represented as the length (the norm) of the vector S:
B=∥S∥=√{square root over (<S,S>)}=√{square root over (D 2 +E 2 +F 2)}  (13)
The Bef color coordinate system is defined as follows:
B=√{square root over (D 2 +E 2 +F 2)}  (14)
e=E/B
f=F/B
If B=0 (absolute black color), then e and f can be left undefined or can be defined in any way, e.g. as zeroes.

Since D is never negative, the D, E, F values can be determined from the B, e, f values as follows:
E=e×B  (15)
F=f×B
D=√{square root over (B 2−(e*B)2−(f*B)2)}{square root over (B 2−(e*B)2−(f*B)2)}=B√{square root over (1−e 2 −f 2)}

The Bef system is convenient for brightness editing because the brightness can be changed by changing the B coordinate and leaving e and f unchanged. Thus, if it is desired to change from some value B to a value k×B for some factor k, the new Bef color coordinates B*, e*, f* can be computed as:
B*=k×B  (16)
e*=e
f*=f
Here k>1 if the brightness is to be increased, and k<1 if the brightness is to be decreased. The transformation (16) corresponds to multiplying the D, E and F values by k:
D*=k×D  (17)
E*=k×E
F*=k×F
This transformation in turn corresponds to multiplying the color's radiance P(λ) by k (see equations (4)). The radiance multiplication by k is a good model for the intuitive understanding of brightness as a measure of the intensity of the color stimulus. Also, there is no color shift in the transformation (16), i.e. the color's chromatic perception does not change.

The brightness can be changed over the whole image or over a part of the image as desired. The brightness transformation can be performed using computer equipment known in the art. For example, in FIG. 4, a computer processor 410 receives the color coordinates B, e, f (from a memory, a network, or some other source) and also receives the factor k (which can be defined by a user via some input device, e.g. a keyboard or a mouse). The processor outputs the k×B, e, f coordinates. In some embodiments, this color editing is controlled by a user who watches the color on a monitor 420 before and after editing. To display the color on monitor 420, the processor 410 may have to convert the color from the Bef coordinate system to some other coordinate system suitable for the monitor 420, e.g. sRGB or some other system as defined by the monitor specification. The color conversion can be performed by converting the color between Bef and DEF as specified by equations (14) and (15), and between DEF and the XYZ system of FIG. 2 by equations (5) and (6). The color conversion between the XYZ coordinate system and the sRGB can be performed using known techniques.

Another color coordinate system that facilitates the brightness editing is the spherical coordinate system for the DEF space. This coordinate system BCH (Brightness, Chroma, Hue) is defined as follows (see also FIG. 3):
B=√{square root over (D 2 +E 2 +F 2)}  (18)

C (“chroma”) is the angle between the color S and the D axis;

H (“hue”) is the angle between (i) the orthogonal projection SEF of the vector S on the EF plane and (ii) the E axis.

The term “chroma” has been used to represent a degree of saturation (see Malacara's book cited above). Recall that in the HSB system, the Saturation coordinate of a color represents the white amount in the color. In the BCH system, the C coordinate of a color is a good representation of saturation (and hence of chroma) because the C coordinate represents how much the color deviates from the white color D65 represented by the D axis (E=F=0).

The H coordinate of the BCH system represents the angle between the projection SEF and the red color represented by the E axis, and thus the H coordinate is a good representation of hue.

Transformation from BCH to DEF can be performed as follows:
D=B×cos C  (19)
E=B×sin cos H
F=B×sin sin H

Transformation from DEF to BCH can be performed as follows. The B coordinate can be computed as in (18). The C and H computations depend on the range of these angles. Any suitable ranges can be chosen. In some embodiments, the angle C is in the range [0,π/2], and hence
C=cos−1(D/B)  (20)
In some embodiments, the angle H can be computed from the relationship:
tan H=F/E  (21A)
or
tan H=f/e  (21B)
where e and f are the Bef coordinates (14). For example, any of the following relations can be used:
H=tan−1(F/E)+α  (22)
H=tan−1(f/e)+α
where α depends on the range of H. In some embodiments, the angle H is measured from the positive direction of the E axis, and is in the range from 0 to 2π or −π to π. In some embodiments, one of the following computations is used:
H=arctg(E,F)  (23)
H=arctg(e,f)
where arctg is computed as shown in pseudocode in Appendix 1 at the end of this section before the claims.

Transformation from BCH to Bef can be performed as follows:
B=B  (24)
e=sin cos H
f=sin sin H
Transformation from Bef to BCH can be performed as follows:
B=B  (25)
C=cos−1√{square root over (1−e 2 −f 2)}, or C=sin−1√{square root over (e 2 +f 2)}
H=tan−1(f/e)+α(see (22)), or H=arctg(e,f) (see Appendix 1).

As with Bef, the BCH system has the property that a change in the B coordinate without changing C and H corresponds to multiplying the tristimulus values D, E, F by some constant k (see equations (17)). The brightness editing is therefore simple to perform. In FIG. 5, the brightness editing is performed like in FIG. 4, with the C and H coordinates used instead of e and f.

The Bef and BCH systems are also suitable for contrast editing. In some embodiments, the contrast is enhanced by increasing the brightness difference between brighter and dimmer image portions. For the sake of illustration, suppose the image consists of a two-dimensional set of pixels pi (i=1, 2, . . . ) with coordinates (xi,yi), as shown in FIG. 6 for an image 610. Let B(pi) denote the brightness value (the B coordinate) at the pixel pi, and B*(pi) denote the modified brightness to be computed to edit the contrast. Let Bavg(pi) be some average brightness value, for example, the mean of the brightness values in some region R(pi) containing the pixel pi:

B avg ( p i ) = 1 N ( p i ) p R ( p i ) B ( p ) ( 26 )
where N(pi) is the number of the pixels in the region R(pi). In the Bef system, for each pixel pi, the new coordinates B*(pi), e*(pi), f*(pi) for the contrast adjustment are computed from the original coordinates B(pi), e(pi), f(pi) as follows:
B*(p i)=B avg 1−ε(p iB ε(p i)  (27.1)
e*(p i)=e(p i)  (27.2)
f*(p i)=f(p i)  (27.3)
Here, ε is some positive constant other than 1. Of note, the equation (27.1) implies that:

B * ( p i ) B avg ( p i ) = ( B ( p i ) B avg ( p i ) ) ɛ ( 28 )
Thus, if ε>1, the contrast increases because the brightness range is increased over the image. If ε<1, the contrast decreases.

The contrast adjustment can be performed over pixels pi of a part of the image rather than the entire image. Also, different ε values can be used for different pixels p1. Alternatively, the same ε value can be used for all the pixels.

Similarly, in the BCH system, the B coordinate is computed as in (27.1), and the C and H coordinates are unchanged:
B*(p i)=B avg 1−ε(p iB ε(p i)  (29.1)
C*(p i)=C(p i)  (29.2)
H*(p i)=H(p i)  (29.3)

In some embodiments, the region R(pi) is the entire image. Thus, the value Bavg(pi) is constant, and can be computed only once for all the pixels pi. In other embodiments, different regions R(pi) are used for different pixels. In FIG. 6, the region R(pi) is a square centered at the pixel pi and having a side of 5 pixels long (the length measured in pixels can be different along the x and y axes if a different pixel density is present at the two axes). In some embodiments, the region R(pi) is a non-square rectangle centered at pi, or a circle or an ellipse centered at pi. In some embodiments, the regions R(pi) have the same size and the same geometry for different pixels pi, except possibly at the image boundary.

In some embodiments, the regions R(pi)) corresponding to pixels pi at the image boundary have fewer pixels than the regions corresponding to inner pixels p1. In other embodiments, the image is extended (or just the brightness values are extended) to allow the regions R(pi) to have the same size for all the pixels pi. Different extensions are possible, and one extension is shown in FIG. 7 for the square regions R(pi) of FIG. 6. The image 610 has n rows (y=0, . . . ,n−1) and m columns (x=0, . . . ,m−1). The image is extended as follows. First, in each row y, the brightness values are extended by two pixels to the left (x=−1, −2) and two pixels to the right (x=m, m+1). The brightness values are extended by reflection around the original image boundaries, i.e. around the columns x=0 and x=m−1. For example, B(−1,0)=B(1,0), B(m,0)=B(m−2,0), and so on. Then in each column (including the extension columns x=−2, −1, m, m+1), the brightness values are extended by two pixels up and two pixels down using reflection. FIG. 7 shows the pixel's coordinates for each pixel and the brightness value for each extension pixel.

Efficient algorithms exist for calculating the mean values Bavg(pi) for a plurality of adjacent pixels pi for rectangular regions R(pi). One such algorithm is illustrated in pseudocode in Appendix 2 at the end of this section.

FIGS. 8 and 9 are brightness graphs illustrating the brightness before and after editing near a color boundary for one example. For simplicity, the image is assumed to be one-dimensional. Each pixel p has a single coordinate x. The coordinate x is the coordinate of some predetermined point in the pixel, e.g. the left edge of the pixel (the x coordinates are discrete values). The color boundary occurs at some pixel x=xo. The original brightness B(x)=a for pixels x<xo, B(x)=b for x≧xo, where a<b. Each region R(pi)=R(x) is an interval [x−R,x+R] of some constant radius R with the center at the point x.

In FIG. 8, ε>1. If x>xo+R or x<xo−R, the average Bavg(x)=B(x). Hence (see equation (28)), B*(x)=B(x). As x decreases from xo+R towards xo, the average Bavg(x) decreases, so B*(x) increases hyperbolically:
B*(x)=c 1(c 2 ×x+c 3)1−ε  (29A)
where c1, c2, c3 are some constants. When x increases from xo−R towards xo, the average Bavg(x) increases, so B*(x) decreases hyperbolically according to equation (29A), with possibly some other constants c1, c2, c3.

In FIG. 9, ε<1. Again, if x>xo+R or x<xo−R, then B*(x)=B(x). As x decreases from xo+R towards xo, the average Bavg(x) decreases, so B*(x) decreases parabolically according to equation (29A). When x increases from xo−R towards xo, the average Bavg(x) increases, so B*(x) increases parabolically according to equation (29A), with possibly some other constants c1, C2, C3. The B* function does not have to be continuous at x=xo.

In some embodiments, Bavg is a weighted average:

B avg ( p i ) = 1 N ( p i ) p R ( p i ) w ( p , p i ) B ( p ) ( 29 B )
where w(p,pi) are some weights. In some embodiments, each value w(p,pi) is completely defined by (x−xi, y−yi) where (x,y) are the coordinates of the pixel p and (xi,yi) are the coordinates of the pixel pi. In some embodiments, all the weights are positive. In some embodiments, the sum of the weights is 1. The sum of the weights can also differ from 1 to obtain a brightness shift. The brightness shift and the contrast change can thus be performed at the same time, with less computation than would be needed to perform the two operations separately.

In some embodiments, the function

( B ( p i ) B avg ( p i ) ) ɛ
in equation (28) is replaced
with some other predefined non-identity function

f ( B ( p i ) B avg ( p i ) ) .
Thus,

B * ( p i ) = B avg ( p i ) × f ( B ( p i ) B avg ( p i ) ) ( 29 C )
In some embodiments, the function ƒ is strictly increasing. In some embodiments, ƒ(1)=1, but this is not necessary if a brightness shift is desired. The brightness shift and the contrast change can thus be performed at the same time, with less computation than would be needed to perform the two operations separately.

The contrast editing techniques described above can be used to adjust either the global contrast, i.e. when the region R(pi) is the whole image, or the local contrast, when the region R(pi) is only a part of the image. In some embodiments, the region R(pi) contains 10% to 30% of the image in the local contrast adjustments.

The same techniques can be used to change the image sharpness if the region R(pi) is small, e.g. 1% of the image or less. In some embodiments, the region R(pi) is contained in a rectangle of at most 31 pixels by 31 pixels, with the center at pixel pi. Even smaller outer rectangles can be used, e.g. 21×21 pixels, or 11×11 pixels, and other rectangles. The image may contain thousands, millions, or some other number of pixels. In FIG. 10, a computer system (such as shown in FIG. 4 or 5 for example) receives an image editing command at step 1010. In a non-limiting example, the command can be issued by a user via a graphical user interface. The command may specify contrast or sharpness editing. If the command is for contrast editing (step 1020), the computer system sets the size of region R(pi) to a suitable value (e.g. 10%˜30% of the image), or defines the region R(pi) to be a large specific region (e.g. the whole image). If the command is for sharpness editing (step 1030), the computer system sets the size of region R(pi) to a small value (e.g. 1% of the image, or a rectangle of 11 pixels by 9 pixels), or defines the region R(pi) to be a small specific region. At step 1040, an editing algorithm is executed as described above with the region or regions R(pi) as defined at step 1020 or 1030. Step 1040, or steps 1020-1040, can be repeated for different pixels p1. The image processing system can be made simpler and/or smaller due to the same algorithm being used for the contrast and sharpness editing.

The contrast and sharpness editing can also be performed on monochromatic images. These images are formed with a single color of possibly varying intensity. In such images, the brightness B represents the color's intensity. In some embodiments, the brightness B is the only color coordinate. The same techniques can be employed for this case as described above except that the equations (27.2), (27.3), (29.2), (29.3) can be ignored.

In some embodiments, the Bef and BCH systems are used to adjust a shadow portion or a highlight portion of the image (e.g. to deepen the shadow and/or brighten (highlight) a brighter image portion). As in equations (27.2), (27.3), (29.2), (29.3) the e and f coordinates or the C and H coordinates are unchanged. The brightness is modified as follows:
B*(p i)=B(p i)×(B 0 /B avg(p i))ε  (30)
where:

B0 is some pre-defined brightness level, e.g. the maximum brightness over the image;

Bavg(pi) is the mean brightness over a region R(pi) containing the pixel pi;

ε is a pre-defined constant; ε is in the interval (0,1) if it is desired to increase the brightness range near the pixel pi, and thus to sharpen the image near the pixel (good results can be achieved with ε≦½); ε is negative (e.g. in the interval (−1,0)) if it is desired to reduce the sharpness range to blur the image.

The Bef and BCH systems are also convenient for performing hue and saturation transformations. In these transformations, the brightness B is unchanged. The hue transformation corresponds to rotating the color vector S (FIG. 3) around the D axis. The chroma coordinate C is unchanged for the hue transformation. Suppose the color vector rotation is to be performed by an angle ø, i.e. the angle H is changed to H+ø. In the BCH system, this transformation is given by the following equations:
B*=B  (31)
C*=C
H*=H+ø

    • while H*>Hmax do H*=H−2π
    • while H*≦Hmin do H*=H+2π
      The H* computation places the result into the proper range from Hmin to Hmax, e.g. from −π to π or from 0 to 2π. If the H* value is outside the interval, then 2π is added or subtracted until the H* value in the interval is obtained. These additions and subtractions can be replaced with more efficient techniques, such as division modulo 2π, as known in the art. In some embodiments, the “>” sign is replaced by “≧” and “≦” by “<” (depending on whether the point Hmin or Hmax is in the allowable range of the H values).

In the Bef system, the hue transformation is given by the following equations:
B*=B  (32)
e*=e×cos ø+sin ø
f*=f×cos ø−sin ø

A saturation transformation can be specified as the change in the chroma angle C, e.g. multiplying C by some positive constant k. The hue and the brightness are unchanged. Since D is never negative, the C coordinate is at most π/2. The maximum C value Cmax(H) can be a function of the H coordinate. In the BCH system, the saturation transformation can be performed as follows:
B*=B  (33)
C*=k×C; if C*>C max(H) then C=C max(H)
H*=H

In the Bef system, it is convenient to specify the desired saturation transformation by specifying the desired change in the e and f coordinates. The saturation change corresponds to rotating the S vector in the plane perpendicular to the EF plane. The angle H remains unchanged. Hence, the ratio e/f=tan H is unchanged, so the saturation transformation can be specified by specifying the ratio e*/e=f*/f. Denoting this ratio by k, the transformation can be performed as follows:
B*=B  (34)
e*=k×e
f*=k×f
Equations (14), (15) imply that
e 2 +f 2≦1  (35)
Therefore, the values e*, f* may have to be clipped. In some embodiments, this is done as follows:
g=√{square root over ((e*)2+(f*)2)}{square root over ((e*)2+(f*)2)}  (36)

if g>1, then
e*=e/g
f*=f/g

The invention includes systems and methods for color image editing and display. The Bef and BCH color coordinates can be transmitted in a data carrier such as a wireless or wired network link, a computer readable disk, or other types of computer readable media. The invention includes computer instructions that program a computer system to perform the brightness editing and color coordinate system conversions. Some embodiments of the invention use hardwired circuitry instead of, or together with, software programmable circuitry.

In some embodiments, the Bef or BCH color coordinate system can be replaced with their linear transforms, e.g. the coordinates (B+e,e,f) or (2B,2e,2f) can be used instead of (B,e,f). The angle H can be measured from the E axis or some other position. The angle C can also be measured from some other position. The invention is not limited to the order of the coordinates. The invention is not limited to DEF, XYZ, or any other color coordinate system as the initial coordinate system. In some embodiments, the orthonormality conditions (9) or (10) are replaced with quasi-orthonormality conditions, i.e. the equations (9) or (10) hold only approximately. More particularly, CMF's t 1(λ), t 2(λ), t 3(λ) will be called herein quasi-orthonormal with an error at most ε if they satisfy the following conditions:

  • 1. each of

0 t _ 1 ( λ ) t _ 2 ( λ ) λ , 0 t _ 1 ( λ ) t _ 3 ( λ ) λ , 0 t _ 2 ( λ ) t _ 3 ( λ ) λ
is in the interval
[−ε, ε], and

  • 2. each of

0 [ t _ 1 ( λ ) ] 2 λ , 0 [ t _ 2 ( λ ) ] 2 λ , 0 [ t _ 3 ( λ ) ] 2 λ
is in the interval [K−ε, K+ε] for positive constants K and ε. In some embodiments, ε is 0.3K, or 0.1K, or some other value at most 0.3K, or some other value. Alternatively, the CMF's will be called quasi-orthonormal with an error at most ε if they satisfy the following conditions:

  • 1. each of

λ t 1 _ ( λ ) t 2 _ ( λ ) , λ t 1 _ ( λ ) t 3 _ ( λ ) , λ t 2 _ ( λ ) t 3 _ ( λ )
is in the interval [−ε, ε], and

  • 2. each of

λ [ t 1 _ ( λ ) ] 2 , λ [ t 2 _ ( λ ) ] 2 , λ [ t 3 _ ( λ ) ] 2
is in the interval [K−ε,K+ε] for positive constants K and ε. In some embodiments, ε is 0.3K, or 0.1K, or some other value at most 0.3K, or some other value. Orthonormal functions are quasi-orthonormal, but the reverse is not always true. If ε=0.1K, the functions will be called 90%-orthonormal. More generally, the functions will be called n%-orthonormal if ε is (100−n)% of K. For example, for 70%-orthonormality, ε=0.3K.

The invention is not limited to the orthonormal or quasi-orthonormal CMF's or to any particular white color representation. For example, in some embodiments, the following color coordinate system (S1,S2,S3) is used for color editing:
S 1=√{square root over (X 2 +Y 2 +Z 2)}  (37)
S 2 =X/S 1
S 3 =Z/S 1
The XYZ tristimulus values in (37) can be replaced with linear RGB tristimulus values or with some other tristimulus values T1, T2, T3, e.g.:
S 1=√{square root over (T 1 2 +T 2 2 +T 3 2)}  (38)
S 2 =T 2 /S 1
S 3 =T 3 /S 1.
If T1 can be negative, than the sign of T1 can be provided in addition to the coordinates. Alternatively the sign of T1 can be incorporated into one of the S1, S2, and/or S3 values, for example:
S 1=Sign(T 1)×√{square root over (T 1 2 +T 2 2 +T 3 2)}  (39)
The brightness editing can still be performed by multiplying the S1 coordinate by k, with the S2 and S3 values being unchanged.

The BCH coordinate system can be constructed from linear systems other than DEF, including orthonormal and non-orthonormal, normalized and non-normalized systems. In some embodiments, a coordinate system is used with coordinates (S1,S2,S3), where S1 is defined as in (38) or (39), and the coordinates S2 and S3 are defined in a way similar to (20)-(23), e.g.:
S 2=cos−1(T 1 /B)  (40)
S 3=tan−1(T 3 /T 2)+α
where α is as in (22), or
S 3=arctg(T 2 ,T 3)  (41)

In some embodiments, the value B is the square root of some other quadratic polynomial of in T1, T2, T3:
B=√{square root over (g 11 T 1 2 +g 22 T 2 2 +g 33 T 3 2 +g 12 T 1 T 2 +g 13 T 1 T 3 +g 23 T 2 T 3)}
wherein g11, g22, g33, g12, g13, g23 are predefined constants, and g11, g22, g33 are not equal to zero (e.g. g11, g22, g33 are positive). Linear transforms result in different mathematical expressions of the coordinates without necessarily changing the values of the coordinates. For example, the coordinate system may use coordinates S1, S2, S3 or their linear transform, wherein S1 is defined by the value B, or S1 is defined by B and the sign of a predefined function of one or more of T1, T2, T3. In some embodiments
B=√{square root over (α1 2(T 1 ,T 2 ,T 3)+α2 2(T 1 ,T 2 ,T 3)+α3 2(T 1 ,T 2 ,T 3))}{square root over (α1 2(T 1 ,T 2 ,T 3)+α2 2(T 1 ,T 2 ,T 3)+α3 2(T 1 ,T 2 ,T 3))}{square root over (α1 2(T 1 ,T 2 ,T 3)+α2 2(T 1 ,T 2 ,T 3)+α3 2(T 1 ,T 2 ,T 3))}
wherein
α1(T 1 ,T 2 ,T 3)=α11 ×T 1 12 ×T 213 ×T 3
α2(T 1 ,T 2 ,T 3)=α21 ×T 1 22 ×T 223 ×T 3
α3(T 1 ,T 2 ,T 3)=α31 ×T 1 32 ×T 233 ×T 3
wherein α11, α12, α13, α21, α22, α23, α31, α32, α33 are predefined numbers such that the following matrix Λ is non-degenerate:

Λ = [ α 11 α 12 α 13 α 21 α 22 α 23 α 31 α 32 α 33 ]
In some embodiments, the coordinate S1 is defined by the value B and by a sign of a predefined function of one or more of T1, T2, T3. The function can be one of α1(T1,T2,T3), α2(T1,T2,T3), α3(T1,T2,T3). Clearly, the values T1′=α1(T1,T2,T3), T2′=α2(T1,T2,T3), T3′=α3(T1,T2,T3) are also tristimulus values. In some embodiments, the coordinate S2 is defined by a value β(T1, T2,T3)/B, wherein
β(T 1 ,T 2 ,T 3)=β1 ×T 12 ×T 23 ×T 3,
wherein β1, β2, β3 are predefined numbers at least one of which is not equal to zero, or the coordinate S2 is defined by the value β(T1,T2,T3)/B and the sign of a predefined function of one or more of T1, T2, T3. The coordinate S3 is defined by a value γ(T1, T2,T3)/B, wherein
γ(T 1 ,T 2 ,T 3)=γ1 ×T 12 ×T 23 ×T 3,
wherein γ1, γ2, γ3 are predefined numbers at least one of which is not equal to zero, and γ(T1,T2,T3) is not a multiple of β(T1,T2,T3), or the coordinate S3 is defined by the value γ(T1,T2,T3)/B and the sign of a predefined function of one or more of T1, T2, T3. In some embodiments, a value cos S2 is defined by a value β(T1 ,T2,T 3)/B, wherein
β(T 1 ,T 2 ,T 3)=β1 ×T 12 ×T 23 ×T 3,
wherein β123 are predefined numbers at least one of which is not equal to zero, or the value cos S2 is defined by the value β(T1,T2,T3)/B and the sign of a predefined function of one or more of T1, T2, T3. In some embodiments, a value tan S3 is defined by a value γ(T1,T2,T3)/δ(T1,T2,T3), wherein
γ(T 1 ,T 2 ,T 3)=γ1 ×T 12 ×T 23 ×T 3,
δ(T 1 ,T 2 ,T 3)=δ1 ×T 12 ×T 23 ×T 3,
wherein γ1, γ2, γ3 are predefined numbers at least one of which is not equal to zero, and δ1, δ2, δ3 are predefined numbers at least one of which is not equal to zero, or the value tan S3 is defined by the value γ(T1,T2,T3)/δ(T1,T2,T3) and the sign of a predefined function of one or more of T1, T2, T3. In some embodiments, β(T1,T2,T3) is one of α1(T1,T2,T3), α2(T1,T2,T3), α3(T1,T2,T3); γ(T1,T2,T3) is another one of α1(T1,T2,T3), α2(T1,T2,T3), α3(T1,T2,T3); and δ(T1,T2,T3) is the third one of α1(T1,T2,T3), α2(T1,T2,T3), α3(T1,T2,T3). In some embodiments, any color whose tristimulus values T1,T2,T3 are such that α1(T1,T2,T3) is not zero and α2(T1,T2,T3)=α3(T1,T2,T3)=0, is a white or a shade of gray. This can be D65 or some other white color. In some embodiments, a monochromatic red color of a predefined wavelength has tristimulus values T1, T2, T3 such that α2(T1,T2,T3) is positive and α3(T1,T2,T3)=0. In some embodiments, α1(T1,T2,T3), α2(T1,T2,T3), α3(T1,T2,T3) are tristimulus values corresponding to 70%-orthonormal color matching functions. In some embodiments, up to a constant multiple, Λ is an orthonormal matrix, i.e. ΛΛT=I, where ΛT is a transpose of Λ, and I is the identity matrix, i.e.

I = [ 1 0 0 0 1 0 0 0 1 ]
We will also call the matrix orthonormal if ΛΛT=c×I where c is a non-zero number. If the elements of ΛΛT are each within c×n/100 of the respective elements of a matrix c×I for some c and n, then the matrix Λ will be called (100−n)%-orthonormal. For example, if the elements of ΛΛT are each within 0.3×c of the respective elements of a matrix c×I (i.e. for each element λi,j of matrix ΛΛT, |λi,j−ei,j|≦0.3×c where ei,j is the corresponding element of the matrix c×I), the matrix Λ will be called 70%-orthonormal. In some embodiments, the matrix Λ is 70% or more percent orthonormal (e.g. 90%-orthonormal).

The tristimulus values T1, T2, T3, or the corresponding color matching functions, do not have to be normalized in any way. In some prior art systems, the color matching functions are normalized to require that the integral of each color matching function be equal to 1. In the XYZ system of FIG. 2, there is a normalizing condition X=Y=Z for a pre-defined, standard white color with a constant spectral power distribution function P(λ). The invention is not limited by any such requirements.

The invention is not limited to the two-dimensional images (as in FIG. 6). The invention is applicable to pixel (bitmap) images and to vector graphics. Colors can be specified separately for each pixel, or a color can be specified for an entire object or some other image portion. The invention is not limited to a particular way to specify the brightness transformation or any other transformations. For example, the brightness transformation can be specified by a value to be added to the brightness (B*=B+k) or in some other way. Other embodiments and variations are within the scope of the invention, as defined by the appended claims.

APPENDIX 1
FUNCTION arctg(a, b)
Here is a pseudocode written in a C-like language (C is a programming
language).
inline float arctg(const float a, const float b)
{
float res;
if( FLT_EPSILON * abs(b) < abs(a) )
//FLT_EPSILON is the minimum positive number ε
// such that 1-ε is different from 1 on the computer.
//abs( ) is the absolute value
{
res = tan−1 (b / a) ;
if(a < 0)
{
res += π ;
}
else if (b < 0)
{
res += 2 π ;
};
}
else
{
if(b < 0)
{
res = 3/2 π ;
}
else
{
if (b > 0)
{
res = ½ π ;
}
else
{
res = 0 ;
};
};
};
return res ;
} ;

APPENDIX 2
Calculation of Means Bavg(pi) for rectangular regions R(pi)
void CalculatePartialSum(
float *pB, //InOut
int nLenght, //In
int nRadius //In
)
{
float *Temp = new float [nLenght + 2 * nRadius] ;
int offset = nRadius ;
for (int i = 0; i < nLenght; ++i)
{
Temp[offset++] = pB[i] ;
} ;
int iLeft = nRadius ;
int iRight = nLenght −2 ;
for (i = 0; i < nRadius; ++i)
{
Temp[i] = pB[iLeft−−] ;
Temp[offset++] = pB[iRight−−] ;
} ;
iLeft = 0;
iRight = 2 * nRadius +1;
float Sum = Temp[0] ;
for (i = 1; i < iRight; ++i)
{
Sum += Temp[i] ;
} ;
pB[0] = Sum;
for (i = 1; i < nLenght; ++i)
{
Sum += Temp[iRight++] − Temp[iLeft++] ;
pB[i] = Sum;
}
return;
} ;
void CalculateMeanBrightness(
float *pMeanB, //Out
image *pImage, //In
int nWidth, //In
int nHeight, //In
int nRadiusWidth, //In
int nRadiusHeight //In
)
{
int length = nWidth * nHeight ;
for(int offset = 0; offset < length; ++offset)
{
pMeanB[offset] =
Brightness(pImage[offset]) ;
} ;
offset = 0;
for(int y = 0; y < nHeight; ++y)
{
CalculatePartialSum( pMeanB+offset,
nWidth, nRadiusWidth ) ;
offset += nWidth ;
} ;
float *Temp = new float[nHeight] ;
float kNorm = 1./(2 * nRadiusWidth + 1) /
(2 * nRadiusHeight + 1) ;
for(int x = 0; x < nWidth; ++x)
{
offset = x ;
for(int y = 0; y < nHeight; ++y)
{
Temp[y] = pMeanB[offset] ;
offset += nWidth ;
} ;
CalculatePartialSum( Temp, nHeight,
nRadiusHeight ) ;
offset = x ;
for(y = 0; y < nHeight; ++y)
{
pMeanB[offset] = Temp[y] * kNorm ;
offset += nWidth ;
} ;
} ;
return ;
} ;

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Classifications
U.S. Classification358/1.9, 358/520, 358/518, 345/581, 345/428
International ClassificationG03F3/08, G06K1/00, G09G5/00, H04N1/60, G06T17/00, G06F15/00
Cooperative ClassificationH04N1/6075, H04N1/6058
European ClassificationH04N1/60G, H04N1/60K
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